On Müntz-type formulas related to the Riemann zeta function

Five basic mistakes are found in the Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane and in the domain of the function, when the left-hand side of equation is finite, the right-hand side may be infinite, and vice versa. The Riemann Zeta function equation holds only at the point Re(s) = 1/2(s= a+ \(i b\)) . However, at this point, the Zeta function is infinite, rather than zero, to make Riemann hypothesis untenable. 2. An integral item around the original point of coordinate system was ignored when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1but divergent when Re(s) < 1 .The integral form of Zeta function does not change the divergence of its series form. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x > 0 . At point x = 0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x = 0 , so the formula can not be used. 4. Because the integral lower limit of Zeta function is zero, the integrand function is not uniformly convergent, so integral sign and sum sign can not be exchanged. But Riemann made them interchangeable, resulting in that the integral form of Zeta function is untenable. 5.The formula \(\theta(x)=\sqrt{x} \theta(1 / x)\) of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x > 0 . Because the lower limit of integral in the deduction was x = 0 , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, the analytic property of the original function is destroyed and the Cauchy-Riemann equation can not be satisfied. So they are not the real zeros of strict Riemann Zeta function.

This thesis deals with two properties of arithmetic functions, namely, factorization and independence. In 1984, Rearick pointed out that the ring of arithmetic functions is a unique factorization domain but is not a principal ideal domain and so is not a Euclidean domain. Without the Euclidean algorithm, the problem of factorizing arithmetic functions becomes quite difficult. In the first part of this thesis, we propose a technique of factorizing certain classes of arithmetic functions and prove some results about factorization which are based mainly on the norms of such functions. The technique is a generalization of the original works of Rearick which consists of solving a special system of linear differential equations whose coefficients are polynomials in the function to be factorized. The solutions of this system are proved to be the sought after factors with increasing norms. Examples illustrating the technique are also given. In 1986, Shapiro and Sparer made an extensive study of algebraic independence of Dirichlet series. Since the ring of Dirichlet series is isomorphic to that of arithmetic functions, the study in one setting is then equivalent to the other. Shapiro and Sparer's investigation began with a theorem asserting that Dirichlet series are algebraically independent if their Jacobian does not vanish, which is classical in the case of real-valued functions. Taking the Riemann zeta function as a building block, they discovered that Dirichlet series algebraically dependent on the zeta function can uniquely be represented as power series in the logarithms of zeta function. A number of algebraic dependence results of Dirichlet series were derived as consequences. Shapiro and Sparer then went on to investigate analogous results for formal generalized Dirichlet series. Results in the second part of this thesis either extend or simplify some of Shapiro and Sperer's results. These include, for example, replacing the zeta function by Dirichlet series with completely multiplicative coefficients to obtain similar log-series expansions, dependence of series with infinite support, and dependence of non-units whose norms are relatively prime.

In this article we examine certain formulae applying to the Riemann zeta function in the critical strip 0<Re(s)<1, and by means of the Laplace transform we demonstrate new relations between existing formulae. Müntz has proposed a formula for the Riemann zeta function in the critical strip, which involves an arbitrary function satisfying certain conditions. From this formula it is clear that a resolution of the Riemann hypothesis may hinge on a successful method for dealing with the sum-integral difference. The Euler–MacLaurin summation formula is one such device. Here, we generate new expressions for certain Laplace transforms involving the sum-integral difference. Subsequently, the formulae thus far established are generalized using an arbitrary positive number m, which makes apparent the dependence of classical analysis on formulae and results associated with the exponential function which is characterized by the particular case m=1.

In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

It is shown that the abscissa of convergence for the Dirichlet series ( − 1 ) k ( 1 − 2 1 − s ) k + 1 ζ ( k ) ( s ) {( - 1)^k}{(1 - {2^{1 - s}})^{k + 1}}{\zeta ^{(k)}}(s) is zero, where ζ ( s ) \zeta (s) is the Riemann zeta function. This implies the existence of infinitely many zeros of ζ ′ ( s ) \zeta ’(s) in the critical strip.

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π 2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers. In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζ G (2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.

It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the Number-Theoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. This in turn leads to the question whether certain graphs are Ramanujan. The general idea is to explain the pseudorandom features of certain number-theoretical functions by considering them as observables of a spin chain of statistical mechanics. In an appendix we relate the free energy of that chain to the Lewis Equation of modular theory.

Müntz formula and zero free regions for the Riemann zeta function

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.

There is a well-known formula for the Hurwitz zeta function which implies the functional equation for the Riemann zeta function. We give a new proof of that formula and recover certain integral representations for the Hurwitz and Riemann zeta functions.

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.

In our previous work (Y. Hashimoto, Selberg’s zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268 ), we proposed an upper bound of the logarithmic derivative of Selberg’s zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.

The Euler–Maclaurin–Siegel and Abel–Plana summation formulae for the entire Riemann functional equation to handle the Riemann hypothesis

This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.

Some problems of the number theory are associated with the zeros of special functions, such as the Riemann zeta function ζ(s), Dirichlet L-functions L(s, χ) and others. The Riemann zeta function is the most famous. On the half-plane ℜs > 1, the Riemann zeta function is defined by Dirichlet series ζ(s) = ∑ +∞ n=1 n −s . In 1859, Riemann conjectured that non-trivial zeros of the Riemann zeta function lie on the critical line ℜs = 1 2 . Hardy was the first to prove in 1914 that an infinity of zeros are on the critical line. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line [T, T + H], H = T 0.5+e . In 1984, A. A. Karatsuba proved Selberg’s result for shorter intervals of critical line [T, T + H], H = T 27 82 +e . For arithmetic Dirichlet series satisfying a functional equation of Riemann type but admitting no Euler product expansions, lower bounds of the correct order of magnitude for the number of their zeros on intervals of the critical line ℜs = 1/2 have not been obtained so far. The first to show that the critical line contains abnormally many zeros of an arithmetic Dirichlet series without Euler products was Voronin, who proved in 1980 that interval (0, T] of critical line contains more than cT e 1 20 √ ln ln ln ln T zeros of the Davenport–Heilbronn function. In 1989 A. A. Karatsuba developed a new method for obtaining lower bounds for the number zeros of certain Dirichlet series in intervals of critical line; by using this method, he substantially strengthened Voronin’s result. In 1991 Karatsuba solved (by his 1989 method) the problem of estimating the number zeros of linear combinations of functions which are analogous the Hardy function. In the present paper we prove a theorem similar to the theorem of A. A. Karatsuba (in 1991), but only for almost all intervals of the form (T, T + H), H = Xe , where e is an arbitrary positive number, and X ≤ T ≤ 2X, X > X0(e).